The crossing number of a simple connected graph $G$ is the minimum number of edge crossings of $G$ over all drawing of $G$. It's well known that on the sphere the crossing number of the complete graph $K_5$ is $1$ and on the torus it's $0$.
What is the crossing number of $K_5$ on the tubular neighborhood of a trefoil knot?
My intuition is that it's still $0$, but I'm having trouble showing this.
Gerry's comment is right on the nose. Draw your $K_5$ knot on the torus with zero crossings. Cut your torus down a circle, twist the "torus" (now a cylinder) to a tubular trefoil, then re-glue along the original identification. This adds no additional crossings to your graph embedding, but now you are embedded into the desired space.